Dynamics of delay-differential equations modelling immunology of tumor growth

A phenomenological model of a tumor interacting with the relevant cells of the immune system is proposed and analysed. The model has a simple formulat ion in terms of delay-differential equations (DDEs). The critical time-dela y, for which a destabilising Hopf bifurcation of the relevant fixed point o ccurs, and the conditions on the parameters for such bifurcation are found. The bifurcation occurs for the values of the parameters estimated from rea l data. Local linear analyses of the stability is sufficient to qualitative ly analyse the dynamics for small time-delays. Qualitative analyses justify the assumptions of the model. Typical dynamics for larger time-delay is st udied numerically. (C) 2001 Elsevier Science Ltd. All rights reserved.